When interpreting electrons as particles, the atom is mostly empty. The position of the electron is not known with certainty, but the number and mass of the electrons is and it is very small compared to the size of the entire atom. Volumetrically electrons are point particles and take up no space at all.
When interpreting electrons as quantum waves the term "empty" is largely meaningless.
The Bohr planetary model, which I assume is what you're referring to, is equivalent to Newtonian mechanics. It's simple to understand and a good starting point for high schoolers. It's even still usable for certain types of calculations.
However, like Newtonian mechanics, at a deeper level and according to modern understanding, it's fundamentally wrong.
Instead we use the Schrodinger electron-cloud model. According to which the ~nucleus~ (EDIT I mean atom not nucleus) is not mostly empty.
It's not pedantic to discuss this, any more than it's pedantic to discuss why Newtonian mechanics is less accurate than relativity.
The electron cloud and the nucleus are different things. The electron cloud on it's own is an empty volume except where the electrons are at any given moment.
The article specifically rejects this way of thinking and makes the point that the electrons are not in the cloud. The electrons are the cloud. Thinking of them as little balls whizzing about so fast that it seems cloud-like is wrong.
It’s common to say they behave as a particle and a wave, but it’s equally valid to say they don’t behave as a particle or a wave. It’s a distinct phenomena and thinking in terms of large scale things you’re familiar with is simply misleading.
This article is trying to help laypeople have better intuition about things from the quantum point of view. It's full of evocative prose. Arguing about the definition of the word "full" is completely tangential to the point.
The point of physics is to have a unified system where the same description applies to the widest possible number of phenomena. Calling atoms empty or full isn’t a great description but neutrino’s behavior for example lends itself to calling them empty.
Quantized fields is a much better term. Also collapse of the wave function never happens, only increased entanglement.
The concept of a single particle as a point charge that isn't really interacting with everything around it is similarly flawed. We learn from a very young age to conceptualize that as the building block of matter, and entanglement as something weird. Instead it's the other way around: a single particle with a pure (unentangled) wavefunction is pretty near statistically impossible.
That isn't entirely a thing.
Fields may have high or low energy densities, and we might compare that to the height and frequency of waves on the ocean in explanation. But just as the ocean is violent or becalmed, not "full" or "empty", so are fields neither full or empty.
wouldn't it still be pretty meaningfully rephrased as 'stochastic density[0]' or similar?
I would presume that a nucleus, as a superposition if its respective waves, has a far higher stochastic density than the rest of an atom. Or rephrased, an atom is mostly [near] empty space.
[0] of energy, or mass
This is fair. But the point is that a model which is fundamentally stochastic is more valid than one where particles are continuously moving through some subset of mostly empty space.